Problem: The sum of two numbers is $115$, and their difference is $47$. What are the two numbers?
Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 115}$ ${x-y = 47}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 162 $ $ x = \dfrac{162}{2} $ ${x = 81}$ Now that you know ${x = 81}$ , plug it back into $ {x+y = 115}$ to find $y$ ${(81)}{ + y = 115}$ ${y = 34}$ You can also plug ${x = 81}$ into $ {x-y = 47}$ and get the same answer for $y$ ${(81)}{ - y = 47}$ ${y = 34}$ Therefore, the larger number is $81$, and the smaller number is $34$.